Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of #67361 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#67361 |
| Fitting | C2×C7×C9 ⋊ C2×C3 |
| derived | C7×C9 ⋊ C22×C3 |
| Hall | #4427 ⋊ C22 |
| Sylow | C7 ⋊ C2×AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #67361 |
| commutator, cosocle, hypocenter | C7×C9 | C22×C3 |
| Frattini | C3 | C2×D3×ΓL(1,𝔽8) |
| Fitting | C2×C7×C9 | C2×C3 |
| socle | C2×C3×C7 | C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
44 conjugacy classes — view the full character table.
| maximal subgroup of | #67361⋊C4, #13094211, #13094217, … |
| maximal normal subgroup of | #67361⋊C4, #13094211, #13094217, … |
| maximal quotient of | #13094276, C4×#67361, C4⋊#67361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 76);